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A permutomino is a member of a class of polyominoes that are defined using a pair of permutations of size n+1, where n is the width and height of the bounding box of the permutomino.

The number of convex permutominoes of size n for the first 10 n is

n Number of
1 1
2 4
3 18
4 84
5 394
6 1,836
7 8,468
8 38,632
9 174,426
10 780,156

The 2x2 permutominoes[edit]

There are 4 convex permutominoes of order 2 (shown in green).

Permutominoes 2x2.svg

The clockwise 3x3 permutominoes[edit]

There are 18 convex and 8 concave order-3 polyominoes (shown in green).

Permutominoes 3x3.svg


  • Paolo Boldi; Violetta Lonati, Roberto Radicioni, Massimo Santini (November 29, 2006). The number of convex permutominoes (pdf). Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano.